\4537 


*B    17fl    75b 


BOND  VALUES 


UNUSUAL  COUPON  RATES 


ARTHUR  S.  LITTLE 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/formulaeforobtaiOOIittrich 


FORMULAE 

t 

For  Obtaining  from  Ordinary 
Bond  Tables 

VALUES  FOR  BONDS 

At  Various  Unusual 
Coupon  Rates 

From  3.01%  to  6.50% 


2067 


Devised  and  Publisht  by 

ARTHUR  S.  LITTLE 

303  N.  Fourth  Street,  St.  Louis 


Copyright,  1915,  by  Arthur  S.  Little 


Price  $2.50 


5*1 


Preface. 

Up  to  1902  the  bond  tables  then  in  use  seem  to  hav 
met  all  requirements,  but  during  that  year  the  Odd 
Rate  problem  arose,  *  and  has  been  a  live  question 
ever  since.  Publishers  hav  been  kept  busy  issuing 
enlarged  or  special  editions  providing  for  bonds  at 
4Ji%>  5J^>%,  etc.,  but  it  is  hopelessly  out  of  the 
question  to  furnish  comprehensiv  tables  for  all 
the  odd  rate  bonds  that  exist  today; — much  less  the 
others  that  are  apt  to  come.  For  there  is  said  to 
be  a  growing  tendency  on  the  part  of  cities,  school 
districts,  etc.,  to  sell  new  issues  at  par  net  and  hav 
the  competition  between  buyers  take  place  in  the 
coupon  rate  insted  of  the  price.  This  plan,  in 
spite  of  whatever  inconvenience  it  may  occasion 
bond  dealers,  investors  and  accountants,  is,  in  my 
opinion,  unquestionably  the  wisest  and  best  from 
the  standpoint  of  political  economy  and  municipal 
accounting,  for  a  municipal  corporation  is  not  sup- 
posed to  hav  a  Capital  or  Profit  &  Loss  account, 
t  and  if  the  coupons  really  represent,  or  rather, 
are  equivalent  to,  the  exact  interest  on  the  out- 
standing bonds,  it  is  quite  possible,  in  conjunction 
with  the  Serial  Bond  feature,  skillfully  applied,  to 
provide  for  an  equitable  system  of  taxation  and 
budget-making  that  is  almost  ideal.  Believing, 
therefore,  that  at  no  distant  date  there  will  be 
frequent  issues  of  municipal  bonds  at  such  rates 
as  3.95%,  6.35%,  etc.,  or  even  3.94%  or  6.37%,  I 
hav  devised  the  accompanying  table  of  formulae 
by  which  values  for  any  odd  rate  may  be  redily 
obtaind  by  adding  together  a  certain  number  of 
times  a  certain  number  of  values  found  in  the 
ordinary  tables  in  use. 

These  formulae  may  be  applied  to  any  table, 
regardless  of  how  frequently  the  coupons  mature, 
or  to  what  time  unit  standard  the  rate  of  income 
is  referd. 

As  work  of  this  nature  will  almost  invariably  be 
performd  in  an  office,  the  use  of  an  adding  machine 
is  of  course  contemplated,  in  which  case  it  is  merely 
a  matter  of  adding,  and  multiplying  does  not  enter 
into  the  operation  (except  effectivly). 

*A  large  issue  of  St.  Louis  World's  Fair  bonds  at  S%%,  which 
caused  quite  a  rumpus  in  investment  circles  and  occasiond  the  pub- 
lication of  a  special  table  to  cover  the  case. 


f  "Its  principal  asset  is  its  power  of  confiscating  the  property  of 
its  members  and  others  within  its  limits,  thru  taxation,  to  an  extent 
which  cannot  be  valued,  but  which  is  mesured  by  the  needs,  as  legally 

ascertained,   of  its  members hence   its  balance  sheet  is 

non-existent.     The    highest    function    of    municipal  bookkeeping    is 
the  coordination  of  revenue  and  expenditure,  of  sacrifice  and  service." 

THE  PHILOSOPHY  OF  ACCOUNTS.     Sprague. 

5l!vQ$ft2 


It  is  also  best,  when  practicable,  to  work  with  an 
assistant;  one  party  operating  the  machine  while 
the  other  dictates  the  steps  to  be  taken,  and  it  will 
no  dout  be  found  that  a  fairly  comprehensiv  set 
of  manuscript  tables  can  be  turnd  out  in  this  man- 
ner in  a  surprisingly  short  time.  However,  even 
when  an  adding  machine  is  not  available,  this 
method  is  always  feasible  with  pencil  &  paper,  and 
in  many  cases  at  least  will  prove  easier,  quicker 
and  safer  than  any  other. 

The  mode  of  procedure  is  simplicity  itself,  and  the 
items  in  the  schedule  simply  represent  the  number 
of  times  the  value  for  the  bond  indicated  at  the  hed 
of  the  colum  is  to  be  written,  or  rather  taken. 

The  total  is  always  the  value  desired. 

The  results  are  always  increast  tenfold  or  a  hun- 
dredfold, and  therefore  one  or  two  final  figures  are 
worthless,  and  must  be  rejected.  In  other  words, 
the  interpolated  value  obtaind  is  never  (except  by 
accident)  accurate  to  a  greater  number  of  places 
than  the  tables  that  were  employd.  The  rejected 
figures  may,  however,  and  in  fact  should  be  utilized 
to  correct  to  the  nearest  unit  the  final  figure  that 
is  retaind. 

The  task  of  concocting  these  formulae  did  not 
seem  to  admit  of  any  well  defined  system  and  was 
more  in  the  nature  of  a  gigantic  jigsaw  puzzle, 
and  of  course  proved  very  laborious  and  trying. 
Furthermore,  it  was  necessary  to  produce  results 
in  a  form  as  equally  well  adapted  as  possible  for 
every  type  of  adding  machine  in  use.  Had  there 
been  no  necessity  for  consideration  of  any  other 
machine  than  the  ten=key  type  with  visible  printing, 
many  of  the  formulae  could  have  been  given  in  a 
much  simpler  form. 

Nevertheless,  the  entire  table  has  been  carefully 
verified  twice  by  independent  and  totally  different 
methods.  No  less  conscientious  care  will  be  exer- 
cized in  examining  the  printer's  proofs;  therefore  the 
table  is  believd  to  be  entirely  correct  thruout. 

ARTHUR  S.  LITTLE 
St.    Louis,    August    10,    1915. 


EXAMPLES. 


Find  a  3.65%  basis  for  a  3.65%  bond,  19  years  to  run. 


3H%  Bond  taken  9  times. 


9795  74 
9795  74 
9795  74 
9795  74 
9795  74 
9795  74 
9795  74 
9795  74 
9795  74 


5%  Bond  taken  1  time. 


....11838  37 
100000  03 


This  problem  is  of  course  an  idle  one,  and  was 
purposely  selected  in  order  to  obtain  a  glaringly 
correct  result.  The  above  is  representativ  of  the 
uniform  accuracy  that  is  always  obtaind. 


Find  an  0%  basis  for  a  4.77%  annual  bond,  1  year 
to  run. 


3M%  bond  2  times. 


/1035  00 
\1035  00 


*V2%  bond  80  times. 


10450  00 
10450  00 
10450  00 
10450  00 
10450  00 
10450  00 
10450  00 
10450  00 


5%  bond  8  times. 


fl050  00 
1050  00 
1050  00 
1050  00 
1050  00 
1050  00 
1050  00 
1050  00 


7%  bond  10  times. 


....  10700  00 
104770  00 


Like  the  foregoing,  this  is  an  impractical  problem. 
It  is  interesting,  however,  and  also  suggests  an  ex- 
cellent and  simple  test  of  the  accuracy  of  any  for- 
mula. 


Find  a  4%  basis  for  a  4.37%  bond,  6  months  to  run. 


3%  bond  9  times. 


4M%  bond,  90  times. 


5%  bond,  1  time. 


Result,  cut  down 


f9950  98 
9950  98 
9950  98 
9950  98 
^9950  98 
9950  98 
9950  98 
9950  98 
9950  98 

100245  10 

100245  10 

100245  10 

100245  10 

100245  10 

100245  10 

100245  10 

100245  10 

100245  10 


10049  02 

1001813  74 
100.1814 


PROOF. 

Add  6  months'  interest  at  4%. 


2.0036 


Value  of  matured  bond  &  coupon 102. 1850 


Find  a  6%  basis  for  a  AhA%  bond,  10  years  to  run. 


3^%  bond,  1  time 8140  32 

f 88841  90 
88841  90 
88841  90 
88841  90 
88841  90 
88841  90 
88841  90 
88841  90 
88841  90 


4^%  bond,  90  times. 


6%  bond,  9  times 90000  00 

897717  42 
Which  cuts  down  to 89.7717 

PROOF 
Add  6  months'  interest  at  6%.  . 2.6932 


92.4649 
Subtract  coupon 2.3125 


Derived  value  for  9H  years 90. 1524 


agreeing  with  result  obtaind  directly,  as 
shown  below. 


8209  52 

89257  10 

89257  10 

89257  10 

89257  10 

89257  10 

89257  10 

89257  10 

89257  10 

89257  10 

90000  00 
901523  42 


SUPPLEMENTARY  REMARKS 

Altho  the  method  of  interpolation  provided  for 
by  the  accompanying  formulae  is  believd  to  be 
thoroly  original  and  superior  to  any  other  yet  de- 
vised for  the  purpose,  at  the  same  time  all  the 
elements  of  novelty  are  strictly  mechanical  in  their 
nature,  and  the  groundwork  of  the  system  is  the 
abstract  general  law  about  to  be  noted.  A  mere 
knowledge  of  the  existence  of  this  relationship  that 
bond  values  at  different  coupon  rates  bear  to  each 
other  is  unfortunately  much  less  widespred  among 
bond  dealers  and  investors  than  it  should  be,  and 
still  less  prevalent  is  an  intelligent  comprehension 
of  the  significance  of  this  law,  and  the  various  ways 
in  which  it  miy  be  invoked  in  the  solution  of  sundry 
practical  problems;  hence  it  is  deemd  worth  while 
to  explain  briefly  why  all  these  different  bond  values 
are  governd  by  this  law. 

If  we  take,  say,  the  3%  income  line  on  the  10 
year  page  of  Rollins'  or  Deguhee's  4«place  bond 
tables  and  compare  the  values  given  we  will  find 
that  they  differ  from  each  other  as  follows: 


7%  Bond 

134.34 

8.59 

6%  Bond 

125.75 

8.58 

5%  Bond 

117.17 

8.59 

4%  Bond 

108.58 

8.58 

3%  Bond     100.00 

Further  similar  experiments  continued  indefi- 
nitly  for  any  rate  of  income  or  any  time  to  run 
always  produce  results  of  the  same  character.  This 
is  highly  suggestiv,  but  is  not  proof.  Better  still, 
it  may  be  an  approximation  that  holds  good  for 
short  values  only.  We  therefore  try  Deguhee's 
6-place  table,  obtaining: 

134.3373 


125.7530 
117.1686 
108.5843 
100.0000 


8.5843 
8.5844 
8.5843 
8.5843 


and  finally  Sprague's  8-place  table: 
134.337278 


125.752958 
117.168639 
108.584319 
100.000000 


8.584320 
8.584319 
8.584320 
8.584319 


Results  of  a  similar  nature  are  yielded  by: 

The  tables  of  the  writer  for  annual  bonds  on 

a  semi-annual  basis; 

The  tables  of  Rollins  for  annual  bonds  on  an 

annual  basis; 

The  low  rate  tables  of  Sprague  for  quarterly 

bonds  on  a  semi-annual  basis; 

The  tables  of  various  authors  for  quarterly 

bonds  on   a  quarterly   basis. 

In  view  of  all  of  this  it  is  impossible  to  do  else  than 
conclude  that  the  bond  tables  in  use  are  governd 
by  a  very  simple  and  inflexible  law. 

Inasmuch  as  we  have  just  observd  that  by  starting 
with  the  value  for  a  3%  bond  and  successivly  adding 
8.584320  we  obtaind  values  for  bonds  at  4%,  5%,  6% 
&  7%,  we  may  safely  conclude  that  this  process  may 
be  continued  indefinitly,  thus  suggesting  at  once 
that  any  bond  table  contains  (potentially)  values 
for  bonds  at  such  coupon  rates  as  8%,  9%,  10%,  etc. 
It  is  also  evident  that  insted  of  starting  with  the 
3%  bond  and  building  up  by  addition  we  may  start 
with  the  7%  bond  and  tear  down  by  subtraction. 
This  process,  after  passing  the  scope  of  the  tables 
produces: 

2%  Bond     9  14  15  6  8  0 
1%  Bond     8  2  8  3   13  6  0 

and  finally 
0%  Bond     74247040 

A  10  year  s.  a.  bond  carrying  coupons  for  0%  is 
manifestly  nothing  more  than  a  promis  to  pay  a 
single  sum  10  years  hence,  and  the  present  worth 
of  Unity,  according  to  the  tables  of  Reussner,  is 
742470418,  tallying  exactly  with  the  value  for 
an  0%  bond  just  obtaind. 

Reussner's  tables  also  contain  a  list  of  values 
known  as  the  Present  Worth  of  an  Annuity  of  Unity 
per  half-year.  For  instance,  at  the  3%  rate,  10 
years,  the  value  given  is  17.  168638785.  This 
means  that;  placing  upon  money  a  value  of  3%, 


compounded  semi-annually,  then  a  salary,  rental, 
etc.,  of  Unity  per  6  months  for  10  years  (total  face 
value  20)  is  worth  today  17.  1686  etc. 

The  present  worth  of  %  of  Unity  is  of  course  J^ 
of  the  above,  or  8. 5843  1939  2,  which  corresponds 
exactly  with  the  constant  difference  that  we  found 
to  exist  between  the  3%,  4%,  5%,  etc.,  bonds. 

The  facts  in  the  case,  therefore,  are  as  follows, 
despite  what  popular  misapprehensions  or  per- 
verted conceptions  may  exist: 

A  bond  is  a  promis  to  pay  a  single  large  sum  at  a 
fixt  time,  accompanied  by  a  chain  of  promises  to  pay 
smaller  sums  *  for  uniform  amounts  at  fixt  times. 

The  par  value  of  the  bond  is  the  aggregate  face 
value  of  these  numerous  promises  to  pay  fixt  sums 
at  specified  times.  For  example,  the  par  value  of 
a  20  year  5%  bond  is  200. 

Investment  in  a  bond  for  gain  consists  in  dis- 
counting, at  some  rate  of  interest,  compounded 
with  some  standard  of  frequency,  these  various 
promises  to  pay  fixt  sums  at  specified  times. 

Or,  putting  it  another  way,  an  investment  for 
gain  is  merely  the  old  story  of  buying  cheap  and 
selling  dear;  the  purchase  of  a  20  year  5%  bond  at 
105^  being  simply  a  case  of  a  man  paying  $1051 .25 
for  a  stock  of  goods  that  he  knows  with  absolute 
certainty  he  will  retail,  during  the  next  20  years,  for 
exactly  $2,000. 

This  unorthdox  but  correct  PAR  just  defined  also 
constitutes  the  critical  or  absolute  value  of  any  bond, 
viz,  an  0%  basis;  being  the  extreme  price  that  an 
investor  may  pay  without  incurring  actual  loss.f 
Or,  it  may  be  said  to  be  the  bond  value  representing 
an  investment  not  for  gain,  but  for  investment's 


♦Usually,  but  not  necessarily  so.  A  bond  having  a  coupon  rate  of 
300%  could,  under  certain  conditions,  be  a  particularly  desirable 
Bnd  expedient  form  of  a  loan,  and  also  constitute,  at  the  proper  price, 
an  exceptionally  suitable  investment  for  certain  classes  of  investors. 
It    would,    however,    be   necessary; 

(a)  For  the  investor  to  regard  the  price  paid  as  a  basic  investment 
value  and  not  a  terrifying,  heart-rending,  soul-racking  "premium" 
that  will  be  "lost." 

(b)  That  both  the  issuing  corporation  and  the  investor  keep  their 
books  along  different  lines  from  the  kindergarten  methods  almost 
universally  practist  at  present. 


t  What  is  commonly  known  as  the  "par"  of  a  bond  is  a  PUNCTUAL 
INTEREST  BASIS;  a  particular  and  unique  form  of  bond  value, 
whose  relations  to  the  countless  other  basic  values  that  exist  bears  a 
striking  analogy  to  the  relations  of  a  circle  to  ellipses  in  general.  This 
Punctual  Interest  Basis  is  far  from  being  merely  a  normal,  natural, 

logical  value  for  a  bond one  that  every  one  knows one 

that  requires  no  skill  to  be  ascertaind,  etc.,  but  on  the  contrary  pos- 
sesses features  of  considerable  importance  in  filosofical  research, 
political  economy,  the  new  school  of  investment  accounting,  actuarial 
computations,  etc.,  which  features,  however,  cannot  be  discust  in 
detail  here. 


sake  alone.  This  of  course  rarely  occurs  in  bonds, 
except  negativly,  when  the  owners  wilfully  abstain 
from  collecting  them  when  they  become  due.  An 
enormous  amount  of  money  is  lockt  up  in  matured 
United  States  bonds  &  coupons  and  pension  checks 
in  this  manner.  But  investment  for  investment's 
own  sake  is  practist  in  numerous  ways  by  people 
in  all  walks  of  life;  favorite  vehicles  being  Post 
Office  or  Express  money  orders,  bank  drafts,  car 
tickets,    postage    stamps,    etc. 

The  bond  values  in  ordinary  use  are  merely  the 
present  worths  of: 

{The  future  payment  of  the  single  sum  re- 
presented by  the  face  of  the  bond, 
The  symmetrical   series  of   payments   repre- 
sented by  the  coupons. 

Hence  any  bond  table,  insted  of  being  a  scale  of 
"prices"  is  merely  a  complex  set  of  present  worth 
tables,  especially  adapted  for  obtaining  conveniently 
and  expeditiously  the  aggregate  present  worth  of 
the  various  promises  to  pay  that,  taken  together, 
constitute   bonds  as  ordinarily  constructed. 

It  will  be  seen,  therefore,  that  while  the  utility 
of  a  bond  table  is  greatly  enhanced  when  values  for 
various  coupon  rates  are  given,  at  the  same  time 
this  feature  is  a  luxury  rather  than  an  essential, 
and  all  that  the  unskilld  layman  absolutely  needs, 
in  order  to  get  anything  he  wants,  are: 

Tables  of  the  Present  Worth  of  Unity  at  vari- 
ous times  &  rates; 

Tables  of  the  Present  Worth  of  an  Annuity 
of  Unity  for  various  times  &  rates. 


3% 

31%  4% 

4|% 

5% 

6% 

7% 

3.01 

99 

1 

3.02 

99 

1 

3.03 

99 

1 

3.04 

99 

1 

3.05 

9 

1 

3.06 

90 

9 

1 

3.07 

90 

8 

2 

3.08 

90 

8 

2 

3.09 

90 

2     8 

3.10 

9 

1 

3.11 

90 

9 

1 

3.12 

90 

9 

1 

3^ 

90 

8 

1 

1 

3.13 

90 

9 

1 

3.14 

90 

8 

2 

3.15 

9 

1 

3.16 

90 

8 

2 

3.17 

90 

3 

7 

3.18 

90 

2 

8 

3.19 

90 

2 

8 

3.20 

9 

1 

3.21 

90 

9 

1 

3.22 

90 

9 

1 

3.23 

90 

7 

3 

3.24 

90 

6 

4 

3.25 

90 

2 

8 

3% 

3J% 

4%  4|%  5%  6% 

7% 

3.26 

90 

2 

8 

3.27 

91 

9 

3.28 

90 

1 

9 

3.29 

90 

9 

3.30 

9 

1 

3.31 

90 

9 

1 

3.32 

90 

8 

2 

3.33 

90 

2 

8 

3.34 

90 

2 

8 

3.35 

3 

7 

3.36 

91 

9 

3.37 

90 

1 

9 

Ws 

90 

9 

3.38 

90 

9 

3.39 

90 

1 

9 

3.40 

2 

8 

3.41 

80 

9 

10 

3.42 

80 

10 

8 

2 

3.43 

80 

9 

10 

1 

3.44 

80 

1 

10 

3.45 

1 

9 

3.46 

9 

90 

1 

3.47 

9 

90 

3.48 

9 

90 

1 

3.49 

9 

90 

1 

3.51 

99 

1 

3% 

3*% 

4% 

4|% 

5% 

6% 

7% 

3.52 

3 

90 

7 

3.53 

2 

90 

8 

3.54 

1 

90 

9 

3.55 

9 

1 

3.56 

90 

9 

1 

3.57 

90 

9 

1 

3.58 

90 

9 

1 

3.59 

91 

9 

3.60 

9 

1 

3.61 

90 

7 

3 

3.62 

90 

3 

7 

3^ 

90 

2 

1 

7 

3.63 

1 

90 

9 

3.64 

90 

1 

9 

3.65 

9 

1 

3.66 

90 

9 

1 

3.67 

90 

9 

1 

3.68 

90 

7 

3 

3.69 

2 

90 

8 

3.70 

8 

2 

3.71 

90 

7 

3 

3.72 

1 

90 

9 

3.73 

90 

1 

9 

3.74 

90 

1 

9 

3.75 

9 

1 

3.76 

90 

9 

1 

3% 

3  2"% 

4% 

4|% 

5% 

6% 

7% 

3.77 

90 

8 

2 

3.78 

90 

7 

3 

3.79 

90 

6 

4 

3.80 

2 

8 

3.81 

1 

90 

9 

3.82 

90 

1 

9 

3.83 

90 

1 

9 

3.84 

90 

1 

9 

3.85 

3 

7 

3.86 

10 

9 

80 

1 

3.87 

9 

10 

80 

1 

3K 

10 

9 

80 

1 

3.88 

9 

10 

80 

1 

3.89 

9 

10 

80 

1 

3.90 

1 

9 

3.91 

9 

91 

3.93 

9 

90 

1 

3.93 

9 

90 

1 

3.94 

9 

90 

3.95 

1 

9 

3.96 

9 

90 

1 

3.97 

8 

90 

2 

3.98 

7 

90 

3 

3.99 

1 

99 

4.01 

99 

1 

4.02 

... 

99   1 

1 

3% 

3i% 

4%  4J%  5% 

6% 

7% 

4.03 

99 

1 

4.04 

1 

90 

9 

4.05 

9 

1 

4.06 

2 

90 

8 

4.07 

90 

6     4 

4.08 

1 

90 

9 

4.09 

91 

9 

4.10 

9 

1 

4.11 

90 

9 

1 

4.12 

90 

9 

1 

IV* 

2 

90 

1 

7 

4.13 

2 

90 

1 

7 

4.14 

2 

90 

8 

4.15 

7 

3 

4.16 

90 

2    1 

7 

4.17 

1 

90 

9 

4.18 

91 

9 

4.19 

90 

1 

9 

4.20 

9 

1 

4.21 

90 

9 

1 

4.22 

90 

8 

2 

4.23 

90 

7 

3 

4.24 

90 

6 

4 

4.25 

7 

3 

4.26 

1 

90 

9 

4.27 

91 

9 

3% 

3*% 

4% 

4|% 

5% 

6% 

7% 

4.28 

90 

1 

9 

4.29 

90 

1 

9 

4.36 

2 

8 

4.31 

9 

11 

80 

4.32 

9 

10 

80 

1 

4.33 

9 

10 

80 

1 

4.34 

9 

10 

80 

1 

4.35 

1 

9 

4.36 

9 

1 

90 

4.37 

9 

90 

1 

*% 

5 

5 

90 

4.38 

9 

90 

1 

4.39 

9 

90 

1 

4.40 

1 

9 

4.41 

9 

91 

4.42 

3 

7 

90 

4.43 

8 

90 

2 

4.44 

1 

9 

90 

4.45 

1 

9 

4.46 

9 

90 

1 

4.47 

9 

90 

1 

4.48 

9 

90 

1 

4.49 

1 

99 

4.51 

8 

90 

2 

4.52 

3 

90 

7 

4.53 

2 

90 

8 

3% 

n% 

4% 

4|% 

5% 

6%     79 

n 

4.54 

i 
...  i  i 

90 

9 

1 

4.55 

1 

...  1  ... 

9 

1 

4.56 

1 

1    ... 

SO 

9 

1 

4.57 

i 
... 

1 

90 

8 

2 

4.58 

\    ... 

90 

7 

3 

! 

4.59 

1  ... 

90 

6 

4 

i 

4.60 

2 

... 

" 

8 

_ 

4.61 

90 

4 

6 

4.63 

1 

90 

9 

m 

i 

90 

I 
9 

4.63 

1 

90 

9 

4.64 

90 

1 

9 

4.65 

9 

1 

4.66 

90 

9 

1 

4.67 

90 

8 

2 

4.68 

90 

7 

3 

4.69 

90 

6 

4 

4.70 

2 

8 

4.71 

90 

4 

6 

4.72 

90 

3 

7 

4.73 

90 

2 

8 

4.74 

90 

1 

9 

4.75 

9 

1 

4.76 

2 

80 

8 

10 

4.77 

2 

80 

8 

10 

4.78 

2 

80 

8 

10 

) 

3% 

3|% 

4% 

4i% 

5% 

6% 

7% 

4.79 

1 

80 

9 

10 

4.80 

1 

9 

4.81 

9 

1 

90 

4.82 

9 

91 

4.83 

9 

90 

1 

4.84 

9 

90 

1 

4.85 

1 

9 

4.86 

9 

1 

90 

4.87 

8 

2 

90 

m 

5 

5 

90 

4.88 

2 

8 

90 

4.89 

1 

9 

90 

4.90 

1 

9 

4.91 

9 

91 

4.92 

9 

90 

1 

4.93 

9 

90 

1 

4.94 

1 

9 

90 

4.95 

1 

9 

4.96 

8 

92 

4.97 

2 

98 

4.98 

1 

99 

4.99 

1 

99 

5.01 

99 

1 

5.02 

99 

1 

5.03 

2 

1 

90 

7 

5.04 

3 

90 

7 

3% 

3J% 

4% 

4i% 

5% 

6% 

7% 

5.05 

1 

8 

l 

5.06 

2 

90 

8 

5.07 

1 

90 

9 

5.08 

1 

90 

9 

5.09 

91 

9 

5.10 

9 

1 

5.11 

90 

9 

1 

5.12 

2 

90 

8 

5H 

1 

1 

90 

8 

5.13 

2 

90 

8 

5.14 

2 

90 

8 

5.15 

2 

1 

7 

5.16 

1 

90 

9 

5.17 

1 

90 

9 

5.18 

91 

9 

5.19 

90 

1 

9 

5.20 

8 

2 

5.21 

3 

80 

7 

10 

5.22 

4 

80 

6 

10 

5.23 

2 

80 

9 

9 

5.24 

3 

80 

7 

10 

5.25 

3 

7 

5.26 

2 

80 

8 

10 

5.27 

1 

80 

9 

10 

5.28 

1 

80 

9 

10 

5.29 

1 

20 

9 

70 

3% 

3|% 

4% 

4*%  5% 

6%  7% 

5.30 

7 

3 

5.31 

20 

9 

1 

70 

5.33 

2 

20 

8 

70 

5.33 

20 

9 

70 

5.34 

20 

2 

8 

70 

5.35 

2 

1 

7 

5.36 

20 

8 

2 

70 

5.37 

20 

3 

7 

70 

Z% 

20 

5 

5 

70 

5.38 

1 

20 

9 

70 

5.39 

20 

1 

9 

70 

5.40 

2 

8 

5.41 

9 

1 

2C 

1 

70 

5.42 

2 

20 

8 

70 

5.43 

2 

20 

8 

70 

5.44 

1 

20 

9 

70 

5.45 

1 

1 

8 

5.46 

9 

10 

1 

80 

5.47 

9 

10 

1 

80 

5.48 

1 

20 

9 

70 

5.49 

9 

10 

80 

5.50 

2 

8 

5.51 

9 

1 

20 

70 

5.52 

10 

8 

2 

80 

5.53 

10 

9 

80 

1 

5.54 

1 

10 

9 

80 

3% 

s*% 

4% 

4§% 

5% 

6%  7% 

5.55 

... 

3 

7 

5.56 

10 

» 

1 

80 

5.57 

10 

8 

' 

2 

80 

5.58 

... 

10 

* 

3 

80 

5.59 

i 

9 

20 

70   ... 

5.60 

2 

' 

8 

5.61 

10 

8 

80 

2 

5.62 

2 

10 

8 

80 

55A 

5 

10 

5 

80 

5.63 

2 

10 

8 

.  .  . 

80 

5.64 

1 

10 

9 

80 

5.65 

1 

1 

8 

5.66 

2 

10 

8 

80 

5.67 

2 

10 

8 

80 

... 

5.68 

1 

10 

9 

80 

5.69 

4 

10 

6 

80 

5.70 

1 

9 

5.71 

9 

1 

90 

5.72 

9 

1 

90 

5.73 

9 

91 

5.74 

8 

2 

SO 

5.75 

1 

9 

5.76 

8 

2 

90 

5.77 

3 

7 

90 

... 

5.78 

8 

90 

2 

5.79 

1 

9 

90 

3% 

34% 

4%    4*% 

5% 

6% 

7% 

5.80 

... 

l 

9 

5.81 

... 

9 

1 

90 

5.82 

... 

9 

91 

5.83 

9 

90 

1 

5.84 

3 

7 

90 

5.85 

1 

L 

9 

5.86 

i 

J 

2 

n 

5.87 

2 

8 

90 

5% 

i 

5 

90 

5.88 

1 

9 

90 

5.89 

1 

9 

90 

5.90 

1 

9 

5.91 

9 

91 

5.92 

9 

90 

1 

5.93 

1 

7 

90 

2 

5.94 

8 

90 

2 

5.95 

3 

7 

5.96 

4 

90 

6 

5.97 

1 

99 

5.98 

1 

99 

5.09 

1 

99 

6.01 

99 

1 

6.02 

2 

90 

8 

6.03 

2 

90 

8 

6.04 

3 

90 

7 

6.05 

2 

J 

L 

7 

3% 

3|% 

4% 

4|% 

5% 

6% 

7% 

6.06 

2 

90 

8 

6.07 

1 

90 

9 

6.08 

1 

90 

9 

6.09 

91 

9 

6.10 

3 

7 

6.11 

9 

1 

20 

70 

6.12 

9 

20 

1 

70 

6^ 

5 

5 

20 

70 

6.13 

9 

20 

1 

70 

6.14 

2 

8 

20 

70 

6.15 

2 

1 

7 

6.16 

20 

2 

8 

70 

6.17 

1 

20 

9 

70 

6.18 

20 

1 

9 

70 

6.19 

20 

1 

9 

70 

6.20 

2 

8 

6.21 

20 

9 

1 

70 

6.22 

20 

8 

2 

70 

6.23 

9 

20 

1 

70 

6.24 

8 

20 

2 

70 

6.25 

3 

7 

6.26 

9 

1 

20 

70 

6.27 

20 

2 

8 

70 

6.28 

20 

9 

70 

6.29 

1 

20 

9 

70 

6.30 

2 

8 

3% 

Si% 

4% 

4|% 

5% 

6% 

7% 

6.31 

10 

9 

l 

80 

6.32 

10 

9 

l 

80 

6.33 

2 

10 

8 

80 

6.34 

10 

1 

9 

80 

6.35 

1 

1 

8 

6.36 

10 

9 

1 

80 

6.37 

10 

9 

l 

80 

ey8 

10 

5 

5 

' 

' 

80 

6.38 

9 

10 

1 

80 

6.39 

8 

10 

2 

80 

6.40 

... 

... 

2 

8 

6.41 

9 

1 

10 

80 

6.42 

8 

2 

10 

80 

6.43 

1 

10 

9 

80 

6.44 

1 

9 

10 

80 

6.45 

1 

1 

8 

646 

9 

( 

1 

10 

... 

80 

6.47 

2 

10 

8 

80 

6.4S 

1 

10 

9 

80 

6.49 

1 

9 

10 

80 

6.50 

... 

2 

8 

